A characterization of weighted approximations by the Post-Widder and the Gamma operators

Abstract

The weighted approximation errors of the Post-Widder and the Gamma operators are characterized for functions in L p (0, ∞), 1 p ∞, with a weight x , ∈ R. Direct and strong converse theorems are proved. Two types of characteristics are used-weighted K-functionals of the approximated function itself and the classical fixed step moduli of smoothness taken on a simple modification of it.

A characterization of weighted approximations by the Post–Widder and the Gamma operators, II

“…Let us note that the strong converse estimates of type A are optimal. Here we extend the research of [8], where, as we mentioned, the case γ 0 = γ ∞ is considered. The extension is not trivial and requires a new idea because the strong converse inequalities of type A heavily rely on precise determination of the constants in some inequalities connected with the operators (see Section 2).…”

“…Thus, the ratio ‖w( f − P s f )‖ p /K 2 w ( f, (4s) −1 ) p is bounded between two numbers with ratio less than 6 when s is big enough! Note that Theorem 1.1 in the case γ 0 = γ ∞ reduces to Theorem 1.1 from [8].…”

“…In [8] we have established for f ∈ L p (w)(0, ∞), 1 ≤ p ≤ ∞ and a weight of the type w = χ γ (i.e. γ 0 = γ ∞ = γ ) the equivalence…”

“…Also in [8] the K -functional on the right-hand side of (1.5) was characterized in the terms of the classical fixed-step moduli of smoothness.…”

“…Earlier contributions related to the inequalities in (1.5) (in the case γ 0 = γ ∞ = γ ) are summarized in [6]. There are only few results in the case γ 0 ̸ = γ ∞ .…”